Questions tagged [forcing]
Forcing is a set theoretic method used mainly for proving independence results. For questions about forcing function please use (differential-equations).
462
questions
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1answer
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Proving $\|x=y\|\cdot \|\phi(x)\|\le\|\phi(y)\|$ in Boolean valued models
This question relates to the Boolean algebra approach to forcing.
Fix a complete Boolean algebra $B$. I'm writing $\|\sigma\|$ for the Boolean value of $\sigma$, where $\sigma$ is a sentence of the ...
2
votes
1answer
43 views
Role of Negation in Tarski Truth and Cohen Forcing Definitions
As I am new to Forcing, I would appreciate any help on whether the following is anywhere near being correct :
Given a Structure M, Enderton, 2001, "A Mathematical Introduction to Logic" defines truth ...
1
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1answer
59 views
Are the $\mathsf{HOD}$s preserved by weakly homogeneous forcings?
We saw the following theorem in class:
Let $M$ be a transitive model of $\mathsf{ZFC}$, let $\Bbb P\in M$ be a weakly homogeneous partially ordered set, let $G$ be $\Bbb P$-generic over $M$ and let ...
1
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1answer
108 views
Forcing with restricted condition: it's definition [closed]
What does it mean to apply the symbol $\Vdash$ to a condition $q$ restricted to $\xi$:
$q\upharpoonright \xi\Vdash\ldots$
as used on the page 5 above lemma 2.4 here; is this
$\upharpoonright$ there ...
7
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0answers
82 views
Elementary example of Baire spaces whose product is not Baire
It is known that there are Baire spaces $X$ and $Y$ whose product is not Baire, the simplest construction I know is due to Cohen and goes as follow:
Let $S$ be a stationary subset of $\omega_1$, then ...
1
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1answer
49 views
Cohen Set Theory and the Continuum Hypothesis p44 Partial Truth Formulae
In Cohen, Set Theory and the Continuum Hypothesis, page 44 the ability to form Partial Truth Formulae is described :
"We leave as an exercise for the reader the proof of the following fact: For each ...
3
votes
1answer
107 views
In Chow's “beginner's guide to forcing”, why is $\bigcup U$ a function?
I'm reading Timothy Chow's A beginner's guide to forcing (in a quest to finally familiarize myself with "boolean-valued model" forcing), and this passage on page 13 threw me for a loop:
... let $P$ ...
0
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0answers
63 views
Equivalent forcing notions
Let $\mathbb{P}=(P,\leq)$ and $\mathbb{Q}=(Q,\leq)$ be forcing notions, i.e. partial order with a smallest element with the property that there are incompatible elements above each element. Let $h:P\...
2
votes
1answer
62 views
Understanding how to construct a tall narrow tree
I am trying to read the following presentation by Hamkins:
http://jdh.hamkins.org/wp-content/uploads/2017/01/Bonn-Logic-Seminar-2017.pdf
At page 30(34) there is a lemma called "Uniform covering ...
2
votes
1answer
40 views
Weakening of the perfect set property
The perfect set property says that every uncountable set of reals contains a perfect subset.
Now consider the following statement:
P: For every $X\subset\mathbb{R}$, either $X$ or $\mathbb{R}\...
5
votes
1answer
103 views
Shooting a club is Baire
I'm attempting this problem from Kunen:
I'm trying to do it by a direct combinatorial argument. Namely, let $C$ be the set of countable ordinals for which there exist such an $\omega$-chain. I want ...
5
votes
1answer
67 views
If $\omega_1$ is inaccessible to the reals, then every $\mathbf{\Sigma}_2^1$ set has the Baire property
Consider the following proof of the assertion in the title:
Say $A=\{x\in 2^\omega: \phi(x,y)\}$, where $\phi$ is a $\Sigma_2^1$ formula and $y\in 2^\omega$.
By Shoenfield's absoluteness,
$$x\...
4
votes
1answer
80 views
Reals and collapsing posets
Let $M$ be a ctm of $ZFC$, and let $P$ be the poset $\prod_{n<\omega}^{\text{fin}} \text{Coll}(\omega,\aleph_n)$ as computed within $M$.
Let $G$ be $P$-generic over $M$. Let $G^{<n}\times G^{\...
6
votes
2answers
75 views
When to use Iterated Forcing
Iterated forcing is useful for separating cardinal characteristics, and to prove the consistency of Martin's axiom. My question is, what is it about these problems determine that a normal forcing ...
2
votes
1answer
42 views
Finding canonical names for equivalence classes
Given $x,y\subset \omega$, define the equivalence relation
$$x=^*y\iff x\Delta y\text{ is finite.}$$
Let $M$ be a ctm of ZFC and $G$ an $M$-generic filter over $P$=Fn$(\omega\times\omega,2)$. Put $g=\...
2
votes
0answers
68 views
Semi-rigid boolean algebras
A forcing construction I'm trying to do seems to require a complete atomless boolean algebra (used as a forcing poset) that is "semi-rigid" in the sense defined below. I'm wondering if anyone has ...
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0answers
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$ZFC^-$ and transitivity
Why on the page 892 Shelah in his Proper & Improper forcing writes
$N_n$ is a countable model of $ZFC^-$ (so $\in^{N_n}$ is $\in \restriction N_n$ but $N_n$ is not necessarily transitive)
I.e. ...
1
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0answers
85 views
Forcing $\diamondsuit$
I'm trying to show that if $\mathbb{P}$ is the set of partial functions $\omega_1\to\omega_1$ with countable domain, and $M$ is a ctm of $ZFC$, then $M[G]\models\diamondsuit$.
I came across this ...
0
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0answers
47 views
Preserving Fat Diamond principle
I recall the definition of a fat diamond principle at $\kappa$.
Definition: $S\subseteq\kappa$ is called fat stationary if $S$ is stationary and for every club $C\subseteq\kappa$ and every $\alpha&...
1
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3answers
189 views
Forcing without models
I´m a bit puzzled by the non-ctm (countable transitive model) approach to forcing as described in Kunen (see here). Kunen calls this "forcing over the universe". I can see how the proofs of (a), (b) ...
4
votes
1answer
129 views
Forcing in sheaf models of set theory - where do the “generics” disappear to?
I am studying "Sheaves in Geometry and Logic" by Mac Lane & Moerdijk.
In their construction of a topos satisfying $\lnot CH$ they work entirely in the Grothendieck topos of double negation ...
1
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1answer
63 views
the $\Vdash$ relation
There is a lemma in Kunen's book Set theory edition Studies in logic 34:
lemma IV.2.30(5) $$p\Vdash\varphi \text{ iff }\neg \exists q\leq p[q \Vdash\neg\varphi].$$
This equivalence has by elementary ...
1
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1answer
90 views
Intuitive explanation of Cohen's forcing (continuum hypothesis) and how Godel proved CH was consistent within ZFC?
I'm new to set theory and am really struggling to get my head around this. Can anyone give me an intuitive explanation so I can get a general grasp? Cheers
5
votes
2answers
126 views
Is a name a sheaf?
The technique of forcing, in set theory, can be expressed in topos theory as a form of reasoning about sheaves on the notion of forcing, $\mathbb{P}$, equipped with a "double negation" Grothendieck ...
1
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1answer
75 views
Unclear manipulation with forcing conditions
To this question in Kunen's introduction to independence proofs
on the page 379, in the proof of lemma V.7.3 you never mention
$q$
in the proof, only
$p$;
why?
I cannot make sense of ...
3
votes
2answers
101 views
Which classes of subsets are absolute under forcing?
Let $X$ be a ‘definable’ Polish space (in the day-to-day, not necessarily the set-theoretic sense, though possible the latter generalises this). Consider a complexity class $\Gamma(X)$ of subsets of $...
1
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0answers
89 views
What goes wrong shooting a club on $\kappa >\aleph_1$?
I'm reading through Jech's book, and in the section Stationary Sets in Generic Extensions (pages 444 and 445), he remarks that the poset used in shooting a club through a stationary $S\subseteq\...
2
votes
1answer
86 views
Models of $\mathrm{V\neq HOD}$
This is basically a proof verification question. I want to find a model $\mathrm{ZFC+V\neq HOD}$ where $\mathrm{HOD}$ is the class of heriditarilly ordinal definable sets. The idea is to add one Cohen ...
2
votes
1answer
77 views
Finite chain condition - Variation of Martin's Axiom statement
In the following $k$ and $w$ will be cardinal numbers.
Consider the classical statement $MA(k)$:
For any partial order $P$ satisfying the countable chain condition (hereafter $ccc$) and any family ...
1
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1answer
68 views
Existence of disjunction of conditions in countable support iteration of Laver forcing
The part i'm interested in, is the part between the brackets. One side note is that the forcing order is reversed and that's why the disjunction is the greatest lower bound instead of least upper ...
2
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0answers
126 views
Creature forcing/a request for reference
I'm looking for a paper/book/volume treating the forcing with creatures.
I've found one by Roslanowski-Shelah, but this one is too much complicated.
Are all references similarly complicated? Some ...
0
votes
1answer
99 views
intuition for PFA
Kunen (Set theory 2011) says on the page 307:
The Proper Forcing Axiom (PFA) is the assertion that $MA_{\mathbb P}(\aleph_1)$ holds for all proper $\mathbb P$.
My question is, what kind of axiom ...
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0answers
62 views
Relative consistency of ZF with respect to IZF
Is there a forcing argument of this fact? Can anybody point me to the place?
The reason I'm asking is because I was reading Heyting-Valued Models for Intuitionistic Set Theory by R.J. Grayson, yet ...
3
votes
0answers
77 views
Standard fact about iterated forcing
I'm studying Laver's proof of the consistency of Borel's conjecture and he says that the following is a standard fact about iterated forcing:
Define $\mathbb{P}^{\alpha \beta}$ as the set of all ...
1
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1answer
61 views
Proof of MM implies non-stationary ideal on $\aleph_1$ is $\aleph_2$ saturated.
I am trying to understand the proof of thm 37.16 of Jech on page 687.
I don't understand the first 4 lines, why does that suffice to proof the theorem? I don't see how they are related. It sais the ...
0
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1answer
34 views
Existence of generic filters when forcing with a class of conditions.
As the title says, suppose we have a countable transitive model of ZFC like $M$ and suppose $\mathbb{P}$ is a class partial order in $M$. My question is how can we prove that the cardinality of ...
1
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2answers
128 views
Nested models for forcing
Given a model $M$ of ZFC, we can define $\mathbb{P}$-names and generic extensions $M[G]$ in terms of $M$.
But the usual framework for forcing involves an outer model $M\subseteq V$ of ZFC and $M$ is ...
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0answers
35 views
Bounded quantification for Heyting Algebras
Hey could somebody help me to prove the following law for Heyting valued models ( this is corollary 1.18 from Bells Boolean valued models and Independence proofs), which governs the assignment of ...
1
vote
1answer
48 views
Definition of a fusion sequence and proof of the fusion lemma.
I was reading Sacks forcing from Jech's Multiple Forcing and reached the definition of a fusion sequence which said:
Definition. $p \le_n q$ means $p \le q$ and every nth branching point of $q$ is ...
3
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0answers
74 views
What's a good way to get an intuition for making forcing posets?
A while ago, I took a long break from set theory and came back recently. Now, I've been more able to understand concepts which used to be completely foreign to me, for example ultrapowers and mice in ...
1
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1answer
58 views
Question on $[\mathbb{N}]^{\omega}$
I'm reading Fremlin's article about $\mathfrak{p} = \mathfrak{t}$, and in 4B proposition, he shows that $\Vdash_{\mathbb{P}}\,\mathcal{P}(\mathbb{N}) = \mathcal{P}(\mathbb{N})\check{}$, where $\mathbb{...
2
votes
1answer
47 views
Absoluteness of $\Delta_0$ formulas for Boolean-Valued models.
I am stuck on the following lemma from Jech's forcing chapter and need a little bit of help.
Lemma. If $\varphi(x_1, \dots, x_n)$ is a $\Delta_0$ formula, then
$$\varphi(x_1, \dots, x_n) \hspace{...
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1answer
35 views
Existence of a name which has elements with given truth values
Let $\mathcal{A}$ be a complete boolean algebra.
Let $ D $ be a set of $ \mathcal{A}-names $, and $ f:D \to \mathcal{A} $ any function (itself a name).
I wonder if it's possible to construct $ \...
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1answer
83 views
A question on metamathematics of forcing.
This question might be a duplicate but i have found no other questions related to this so here we go.
In kunens (An introduction to independence proofs) and Nik weavers (forcing for mathematicians) ...
2
votes
1answer
59 views
Iterated forcing and non-existance of a model $N$ such that $\langle G_n : n \in \omega \rangle \in N$ and $\text{o}(N) = \text{o}(M)$.
I have encountered the following exercise in kunen's forcing chapter and i have only partially solved it. Any hint or sketch would be very helpful.
Let $\mathbb{P} \in M$ be non-atomic. Let $$M = ...
0
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0answers
110 views
Unique lifts for small forcing extensions (set theory)
I am taking a directed research course in set theory and I am having a lot of trouble, here is the problem I'm still having trouble with:
Problem: Prove that every embedding $j: V \rightarrow M$ in $...
5
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0answers
86 views
If $\mathbb{P}$ is an infinite notion of forcing then there is $H\subset \mathbb{P}$ such that $M[H]$ is not a model of ZFC.
I am trying to prove the following statement and i think i have reached a proof but i'm not sure, is my proof correct or have i missed something?
If $M$ is a countable transitive model of ZFC and $\...
0
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0answers
40 views
Application of forcing to first-order properties of rings
I'm not very well acquainted with forcing, just with the basic ideas; but I thought of the following proof, and since I'm definitely not comfortable with forcing I don't know if it's right, so my ...
3
votes
1answer
81 views
How does Amoeba forcing add a measure 1 set of random reals?
Amoeba forcing is defined as follows:
$\begin{equation*}
(\mathbb{A}, \leq) := (\{X \subset 2^{\omega} : X \text{ is open and } \mu(X) < 1/2\}, \supseteq)
\end{equation*}$.
It is claimed that ...
5
votes
3answers
94 views
The Boolean-valued model $V^B$ is extensional.
I am currently studying forcing from Jechs Set Theory and i have encountered this seemingly innocent lemma which says:$\\$
Lemma 14.17. $V^B$ is extensional.
Proof. Let $X,Y \in V^B$. By ...
